Results 1  10
of
17
The Complexity of McKay's Canonical Labeling Algorithm
, 1996
"... We study the time complexity of McKay's algorithm to compute canonical forms and automorphism groups of graphs. The algorithm is based on a type of backtrack search, and it performs pruning by discovered automorphisms and by hashing partial information of vertex labelings. In practice, the algo ..."
Abstract

Cited by 49 (1 self)
 Add to MetaCart
We study the time complexity of McKay's algorithm to compute canonical forms and automorphism groups of graphs. The algorithm is based on a type of backtrack search, and it performs pruning by discovered automorphisms and by hashing partial information of vertex labelings. In practice, the algorithm is implemented in the nauty package. We obtain colorings of Furer's graphs that allow the algorithm to compute their canonical forms in polynomial time. We then prove an exponential lower bound of the algorithm for connected 3regular graphs of colorclass size 4 using Furer's construction. We conducted experiments with nauty for these graphs. Our experimental results also indicate the same exponential lower bound.
Algebraic Combinatorics in Mathematical Chemistry. Methods and Algorithms. I. Permutation Groups and Coherent (Cellular) Algebras
, 1997
"... ..."
Exact Expectations and Distributions or the Random Assignment Problem
, 1999
"... A generalization of the random assignment problem asks the expected cost of the minimumcost matching of cardinality k in a complete bipartire graph Kmr*, with independent random edge weights. With weights drawn from the exponential(l) distribution, the answer has been conjectured 1 to be ]1,5_ ..."
Abstract

Cited by 20 (0 self)
 Add to MetaCart
A generalization of the random assignment problem asks the expected cost of the minimumcost matching of cardinality k in a complete bipartire graph Kmr*, with independent random edge weights. With weights drawn from the exponential(l) distribution, the answer has been conjectured 1 to be ]1,5_>0, i+5< k (i)('5) ' Here, we prove the conjecture for k < 4, k = rn = 5, and k = rn = n = 6, using a structured, automated proof technique that results in proofs with relatively few cases. The method yields not only the minimum assignment cost's expectation but the Laplace transform of its distribution as well. From the Laplace transform we compute the variance in these cases, and conjecture that with k = rn = n  e<>, the variance is 2/n+ O (log n/n 2 ). We also include some asymptotic properties of the expectation and variance when k is fixed.
Topics in Algebraic Graph Theory
"... The rapidly expanding area of algebraic graph theory uses two different branches of algebra to explore various aspects of graph theory: linear algebra (for spectral theory) and group theory (for studying graph symmetry). These areas have links with other areas of mathematics, such as logic and harmo ..."
Abstract

Cited by 17 (0 self)
 Add to MetaCart
The rapidly expanding area of algebraic graph theory uses two different branches of algebra to explore various aspects of graph theory: linear algebra (for spectral theory) and group theory (for studying graph symmetry). These areas have links with other areas of mathematics, such as logic and harmonic analysis, and are increasingly being used in such areas as computer networks where symmetry is an important feature. Other books cover portions of this material, but this book is unusual in covering both of these aspects and there are no other books with such a wide scope. This book contains ten expository chapters written by acknowledged international experts in the field. Their wellwritten contributions have been carefully edited to enhance readability and to standardize the chapter structure, terminology and notation throughout the book. To help the reader, there is an extensive introductory chapter that covers the basic background material in graph theory, linear algebra and group theory. Each chapter concludes with an extensive list of references.
Architecture of basic building blocks in protein and domain structural interaction networks
, 2005
"... ..."
MOLGEN–CID, A Canonizer for Molecules and Graphs Accessible through the Internet
 J. Chem. Inf. Comput. Sci
, 2004
"... The MOLGEN Chemical Identifier MOLGENCID is a software module freely accessible via the Internet. For a molecule or graph entered in molfile format it produces, by a canonical renumbering procedure, a canonical molfile and a unique character string that is easily compared by computer to a similar s ..."
Abstract

Cited by 10 (7 self)
 Add to MetaCart
The MOLGEN Chemical Identifier MOLGENCID is a software module freely accessible via the Internet. For a molecule or graph entered in molfile format it produces, by a canonical renumbering procedure, a canonical molfile and a unique character string that is easily compared by computer to a similar string. The mode of operation of MOLGENCID is detailed and visualized with examples.
Algorithms for Groups
, 1994
"... Group theory is a particularly fertile field for the design of practical algorithms. Algorithms have been developed across the various branches of the subject and they find wide application. Because of its relative maturity, computational group theory may be used to gain insight into the general str ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
Group theory is a particularly fertile field for the design of practical algorithms. Algorithms have been developed across the various branches of the subject and they find wide application. Because of its relative maturity, computational group theory may be used to gain insight into the general structure of algebraic algorithms. This paper examines the basic ideas behind some of the more important algorithms for finitely presented groups and permutation groups, and surveys recent developments in these fields.
Experimental Studies of the Universal Chemical Key (UCK) Algorithm on the NCI
 Database of Chemical Compounds, Proceedings of the 2003 IEEE Computer Society Bioinformatics Conference (CSB 2003), IEEE Computer Society, Los Alamitos
"... We have developed an algorithm called the Universal Chemical Key (UCK) algorithm that constructs a unique key for a molecular structure. The molecular structures are represented as undirected labeled graphs with the atoms representing the vertices of the graph and the bonds representing the edges. T ..."
Abstract

Cited by 4 (3 self)
 Add to MetaCart
(Show Context)
We have developed an algorithm called the Universal Chemical Key (UCK) algorithm that constructs a unique key for a molecular structure. The molecular structures are represented as undirected labeled graphs with the atoms representing the vertices of the graph and the bonds representing the edges. The algorithm was tested on 236,917 compounds obtained from the National Cancer Institute (NCI) database of chemical compounds. In this paper we present the algorithm, some examples and the experimental results on the NCI database. On the NCI database, the UCK algorithm provided distinct unique keys for chemicals with different molecular structures. 1.
Practical graph isomorphism, ii
 Journal of Symbolic Computation
"... We report the current state of the graph isomorphism problem from the practical point of view. After describing the general principles of the refinementindividualization paradigm and proving its validity, we explain how it is implemented in several of the key programs. In particular, we bring the d ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We report the current state of the graph isomorphism problem from the practical point of view. After describing the general principles of the refinementindividualization paradigm and proving its validity, we explain how it is implemented in several of the key programs. In particular, we bring the description of the best known program nauty up to date and describe an innovative approach called Traces that outperforms the competitors for many difficult graph classes. Detailed comparisons against saucy, Bliss and conauto are presented.
A semicanonical form for sequential AIGs
"... In numerous EDA flows, timeconsuming computations are repeatedly applied to sequential circuits. This motivates developing methods to determine what circuits have been processed already by a tool. This paper proposes an algorithm for semicanonical labeling of nodes in a sequential AIG, allowing pr ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
In numerous EDA flows, timeconsuming computations are repeatedly applied to sequential circuits. This motivates developing methods to determine what circuits have been processed already by a tool. This paper proposes an algorithm for semicanonical labeling of nodes in a sequential AIG, allowing problems or subproblems solved by an EDA tool to be cached with their computed results. This can speed up the tool when applied to designs with isomorphic components or design suites exhibiting substantial structural similarity. 1.